42 The Law of Distribution [CH. m 



At this point, by equation (68), the value of K is 



1 fn\- I fn\ 3 



SW e '"*"3W e '* 



Expanded in powers of e 1} e. 2 , ... this becomes 



i ^ *if e 3 + 



N ' s=1 H ' 



or, since N and are supposed to be infinite, 



For small values of e u e 2 ... the right-hand member of this equation 

 reduces to its first term, which is positive for all values of the e's. It follows 

 that except at the point (72) the value of K is always greater than the value 

 which it has at this point, and therefore that this value is a true minimum 

 value. Moreover, since the point in question is the only point at which 

 8K = 0, it follows that there are no other maxima or minima for K and 

 therefore that K is positive at all points except the point given by (72), 

 at which it vanishes. 



45. On the right-hand of equation (75) the second term stands to the 

 first in the ratio 



(76) 



and the condition that this shall vanish is that the e's shall vanish in com- 

 parison with N a condition which of course admits of the e's being finite or 

 even infinite. If the condition is satisfied, equation (75) may be replaced by 



/ I V M 



2 i 2 I _I_ 2 ^77^ 



W 



If we substitute the values of e l} e 2 ... from equations (73) this becomes 



^JV 



n \n n 



so that K is proportional to the square of the distance from the point at 

 which K = 0. 



If K is kept constant, equation (78) expresses that in the generalised 



N N 



space the distance from the fixed point , ... to the variable point 3d, X 2 ... 



n n 



/ f^N^K\ 

 is constant, and equal to . / ( - - ) . . Equation (78) is therefore the equa- 



tion of a sphere of n dimensions, its radius being . /( ) The centre of 



